Children below age 18 cannot file Form N-400 for naturalization and they need to turn 18 in order to apply for US citizenship. If you are a US citizen, your child will automatically become a US citizen and you just need to file a petition for a citizenship certificate for your child that would help him to prove his citizenship status. Similarly, if you are a naturalized US citizen and if your son was a minor below age 18, when you received your naturalization certificate, your son can acquire citizenship from you and obtain a citizenship certificate, by filing Form N-600, Application for Certificate of Citizenship.
If you are a Green Card holder and if you are applying for US citizenship, your son will become a US citizen at the time you obtain US citizenship. You will have to enter the names of your children in Form N-400, Application for Naturalization and your children below age 18 will become US citizens when you become a naturalized US citizen. And you can apply for US passports for yourself and your children and remain as US citizens, anywhere in the country. But a separate application for naturalization must be filed for your son, if he was not a minor below age 18 when you became a naturalized US citizen and you will have to remember that, Green Card holders above age 18 alone can file applications for naturalization.
Hence, if you and your spouse were both US citizens when your son was born, your son need not apply for US citizenship. He can derive citizenship and he just need to file Form N-600 and obtain a citizenship certificate, to prove his citizenship status. Your son can obtain a citizenship certificate even if you were not a US citizen and if your spouse was a US citizen and vice versa. In this case, your son’s birth certificate must contain the name(s) of his parent(s) who were US citizens, when he was born.
Likewise, if you or your spouse or both became naturalized US citizens, before your son turned 18, he will automatically become a US citizen. As the naturalization laws differ, you will have to be aware of the citizenship laws that were in effect at the time of you son’s birth. Your child cannot automatically become a US citizen, when you become a US citizen, if he is not a permanent resident and if he is an undocumented immigrant. Your son can acquire citizenship from you only after he becomes a lawful permanent resident. If your child does not meet the N-600 requirements, he can apply for US citizenship only after he turns 18, by filing Form N-400.